diff --git a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf
index 70ebb7b..7599732 100644
Binary files a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf and b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf differ
diff --git a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex
index 0929fe5..74f1341 100644
--- a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex
+++ b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex
@@ -599,9 +599,23 @@ $=\displaystyle{\int_0^1\dfrac{1+x}{1+x^2}\textrm{d}x}=\displaystyle{\int_0^1\df
 
 $=\left[\arctan x+\dfrac{1}{2}\ln(1+x^2)\right]_0^1=\dfrac{\pi}{4}+\dfrac{1}{2}\ln2$。
 
-\subsection{变限积分}
+\subsection{变限积分与极限}
 
-\subsection{牛莱公式}
+变限积分也常与极限共同出现。
+
+\textbf{例题：}若$f(x)$连续，$f(0)=0$，$f'(0)=\pi$，求$\lim\limits_{x\to0}\dfrac{\int_0^xf(t)\,\textrm{d}t}{x^2}$。
+
+解：$=\lim\limits_{x\to0}\dfrac{f(x)}{2x}=\dfrac{1}{2}\lim\limits_{x\to0}\dfrac{f(x)-f(0)}{x-0}=\dfrac{1}{2}f'(0)=\dfrac{\pi}{2}$。
+
+\textbf{例题：}若$f(x)$连续，$F(x)=\int_0^x(x-t)f(t)\,\textrm{d}t$，求$F''(x)$。
+
+解：因为$x$与$t$混合在一起很麻烦，$x$为上限是常数，$t$为积分变量。
+
+$F(x)=\int_0^xxf(t)\,\textrm{d}t-\int_0^xtf(t)\,\textrm{d}t=x\int_0^xf(t)\,\textrm{d}t-\int_0^xtf(t)\,\textrm{d}t$
+
+$\therefore F'(x)=\int_0^xf(t)\,\textrm{d}t+xf(x)-xf(x)=\int_0^xf(t)\,\textrm{d}t$。
+
+$\therefore F''(x)=f(x)$。
 
 \subsection{换元积分}
 
diff --git a/advanced-math/knowledge/0-perpare/perpare.pdf b/advanced-math/knowledge/0-perpare/perpare.pdf
index 02517d0..61c6e74 100644
Binary files a/advanced-math/knowledge/0-perpare/perpare.pdf and b/advanced-math/knowledge/0-perpare/perpare.pdf differ
diff --git a/advanced-math/knowledge/0-perpare/perpare.tex b/advanced-math/knowledge/0-perpare/perpare.tex
index 9ca4a55..f1bf6c3 100644
--- a/advanced-math/knowledge/0-perpare/perpare.tex
+++ b/advanced-math/knowledge/0-perpare/perpare.tex
@@ -29,6 +29,8 @@
 % 设置表格行距
 \usepackage{pifont}
 % 圆圈序号
+\usetikzlibrary{arrows.meta}
+\usepackage{scalerel} %\scaleobj{1.5}{} 缩放公式大小
 \author{Didnelpsun}
 \title{考研数学准备}
 \date{}
@@ -53,74 +55,93 @@
 
 \subparagraph{常数函数} \leavevmode \medskip
 
-$y=A$，A为常数，图像平行于x轴：
-
-\begin{tikzpicture}[domain=-1:5]
-    \draw[-latex](-1,0) -- (5,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0, 1.5) node[above]{$y$};
-    \draw[black, thick](-1,1) -- (5,1) node[right]{$y=A$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \filldraw[black] (0,1) circle (2pt) node at(0.75,0.5){$(0,A)$};
-\end{tikzpicture}
+\begin{minipage}{0.35\linewidth}
+    $y=A$，A为常数，图像平行于x轴。
+\end{minipage}
+\hfill
+\begin{minipage}{0.55\linewidth}
+    \begin{tikzpicture}[domain=-1:5]
+        \draw[-latex](-1,0) -- (5,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0, 1.5) node[above]{$y$};
+        \draw[black, thick](-1,1) -- (5,1) node[right]{$y=A$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \filldraw[black] (0,1) circle (2pt) node at(0.75,0.5){$(0,A)$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{幂函数} \leavevmode \medskip
 
-$y=x^{\mu}$，$\mu$为实数，当$x>0$，$y=x^{\mu}$都有定义：
+\begin{minipage}{0.4\linewidth}
+    $y=x^{\mu}$，$\mu$为实数，当$x>0$，$y=x^{\mu}$都有定义：
 
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,4) node[above]{$y$};
-    \draw[black, thick, smooth, domain=0.3:2] plot (\x,1/\x) node[right]{$\mu =-1$};
-    \draw[black, thick, smooth, domain=-2:-0.5] plot (\x,1/\x) node[right]{$\mu =-1$};
-    \draw[black, thick, smooth, domain=0.01:2] plot (\x, {sqrt(\x)}) node[right]{$\mu =\dfrac{1}{2}$};
-    \draw[black, thick, smooth, domain=-2:2] plot (\x,\x) node[right]{$\mu =1$};
-    \draw[black, thick, smooth, domain=-2:2] plot (\x, {\x*\x}) node[right]{$\mu =2$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \filldraw[black] (1,1) circle (2pt) node at(1.25,0.5){$(1,1)$};
-\end{tikzpicture}
+    \begin{tikzpicture}[scale=0.9]
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,4) node[above]{$y$};
+        \draw[black, thick, smooth, domain=0.3:2] plot (\x,1/\x) node[right]{$\mu =-1$};
+        \draw[black, thick, smooth, domain=-2:-0.5] plot (\x,1/\x) node[right]{$\mu =-1$};
+        \draw[black, thick, smooth, domain=0.01:2] plot (\x, {sqrt(\x)});
+        \filldraw[black] (2.75,1.25) node {$\mu =\dfrac{1}{2}$};
+        \draw[black, thick, smooth, domain=-2:2] plot (\x,\x) node[right]{$\mu =1$};
+        \draw[black, thick, smooth, domain=-2:2] plot (\x, {\x*\x}) node[right]{$\mu =2$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \filldraw[black] (1,1) circle (2pt) node at(1.25,0.5){$(1,1)$};
+    \end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}{0.5\linewidth}
+    对于幂函数可以根据不同幂下相同单调性来研究最值：
 
-所以对于幂函数，可以根据不同幂下相同单调性来研究最值：
-
-\begin{enumerate}
-    \item $\sqrt{u}，\sqrt[3]{u}$可以使用$u$来研究。
-    \item $\vert u\vert$可以使用$u^2$来研究。
-    \item $\dfrac{1}{u},u>0$可以使用$u$来研究，但是最值相反。
-    \item $u_1u_2...u_n$可以使用$\sum_{i=1}^{n}\ln u_i$来研究。
-\end{enumerate}
+    \begin{enumerate}
+        \item $\sqrt{u}，\sqrt[3]{u}$可以使用$u$来研究。
+        \item $\vert u\vert$可以使用$u^2$来研究。
+        \item $\dfrac{1}{u},u>0$可以使用$u$来研究，但是最值相反。
+        \item $u_1u_2...u_n$可以使用$\sum_{i=1}^{n}\ln u_i$来研究。
+    \end{enumerate}
+\end{minipage}
 
 \subparagraph{指数函数} \leavevmode \medskip
 
-$y=a^x(a>0,a\neq 1)$：
+\begin{minipage}{0.4\linewidth}
+    $y=a^x(a>0,a\neq 1)$：
 
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
-    \draw[black, thick, domain=-2:2] plot (\x,{pow(1/2,\x)}) node[right]{$0<a<1$};
-    \draw[black, thick, domain=-2:2] plot (\x,{pow(2,\x)}) node[right]{$a>1$};
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
+    \begin{tikzpicture}[scale=0.9]
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
+        \draw[black, thick, domain=-2:2] plot (\x,{pow(1/2,\x)}) node[right]{$0<a<1$};
+        \draw[black, thick, domain=-2:2] plot (\x,{pow(2,\x)}) node[right]{$a>1$};
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}{0.5\linewidth}
+    指数函数具有如下性质：
 
-指数函数具有如下性质：
-
-\begin{enumerate}
-    \item 特殊函数值：$a^0=1$。
-    \item 定义域：$(-\infty, +\infty)$，值域：$(0,+\infty)$。
-    \item 单调性：$a>1$，$y=a^x$单调增，$0<a<1$，$y=a^x$单调减。
-    \item 常用指数函数：$y=e^x$。
-    \item 极限：$\lim\limits_{x\to -\infty}e^x=0$，$\lim\limits_{x\to +\infty}e^x=+\infty$。
-\end{enumerate}
+    \begin{enumerate}
+        \item 特殊函数值：$a^0=1$。
+        \item 定义域：$(-\infty, +\infty)$，值域：$(0,+\infty)$。
+        \item 单调性：$a>1$，$y=a^x$单调增，$0<a<1$，$y=a^x$单调减。
+        \item 常用指数函数：$y=e^x$。
+        \item 极限：$\lim\limits_{x\to -\infty}e^x=0$，$\lim\limits_{x\to +\infty}e^x=+\infty$。
+    \end{enumerate}
+\end{minipage}
 
 \subparagraph{对数函数} \leavevmode \medskip
 
-$y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数：
+\begin{minipage}{0.45\linewidth}
+    $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数：
 
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-0.5,0) -- (4,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \draw[black, thick, domain=0.2:4] plot (\x,{ln(1/\x)}) node[right]{$0<a<1$};
-    \draw[black, thick, domain=0.2:4] plot (\x,{ln(\x)}) node[right]{$a>1$};
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
+    常用公式：$x=e^{\ln x}$，$u^v=e^{\ln u^v}=e^{v\ln u}(x>0,u>0)$。
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    \begin{tikzpicture}[scale=0.9]
+        \draw[-latex](-0.5,0) -- (4,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \draw[black, thick, domain=0.2:4] plot (\x,{ln(1/\x)}) node[right]{$0<a<1$};
+        \draw[black, thick, domain=0.2:4] plot (\x,{ln(\x)}) node[right]{$a>1$};
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
 
 对数函数具有如下性质：
 
@@ -130,9 +151,9 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数：
     \item 单调性：$a>1$，$y=\log_ax$单调增，$0<a<1$，$y=\log_ax$单调减。
     \item 常用对数函数：$y=\ln x$，$e=2.71828...$。
     \item 极限：$\lim\limits_{x\to 0^+}\log_a x=-\infty$，$\lim\limits_{x\to +\infty}\log_ax=+\infty$。
-    \item 常用公式：$x=e^{\ln x}$，$u^v=e^{\ln u^v}=e^{v\ln u}(x>0,u>0)$
 \end{enumerate}
 
+
 \subparagraph{三角函数} \leavevmode \medskip
 
 正弦函数：
@@ -177,40 +198,44 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数：
     \item 有界性：$\vert\sin x\vert\leqslant 1$，$\vert\cos x\vert\leqslant 1$。
 \end{enumerate}
 
-正切函数：
+\begin{minipage}{0.5\linewidth}
+    正切函数：
 
-\begin{tikzpicture}[scale=0.7]
-    \draw[-latex](-6,0) -- (6,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \draw[black, thick, domain=-pi/2+0.5:pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$};
-    \draw[black, densely dashed](pi/2,2) -- (pi/2,-2);
-    \draw[black, densely dashed](-pi/2,2) -- (-pi/2,-2);
-    \draw[black, thick, domain=-pi/2*3+0.5:-pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$};
-    \draw[black, densely dashed](pi/2*3,2) -- (pi/2*3,-2);
-    \draw[black, thick, domain=pi/2+0.5:pi/2*3-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$};
-    \draw[black, densely dashed](-pi/2*3,2) -- (-pi/2*3,-2);
-    \filldraw[black] (0,0) node[below]{$O$};
-    \filldraw[black] (pi/2+0.5,-0.75) node{$\dfrac{\pi}{2}$};
-    \filldraw[black] (-pi/2-0.75,-0.75) node{$-\dfrac{\pi}{2}$};
-    \filldraw[black] (pi/2*3+0.5,-0.75) node{$\dfrac{3\pi}{2}$};
-    \filldraw[black] (-pi/2*3-0.75,-0.75) node{$-\dfrac{3\pi}{2}$};
-\end{tikzpicture}
+    \begin{tikzpicture}[scale=0.55]
+        \draw[-latex](-6,0) -- (6,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \draw[black, thick, domain=-pi/2+0.5:pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$};
+        \draw[black, densely dashed](pi/2,2) -- (pi/2,-2);
+        \draw[black, densely dashed](-pi/2,2) -- (-pi/2,-2);
+        \draw[black, thick, domain=-pi/2*3+0.5:-pi/2-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$};
+        \draw[black, densely dashed](pi/2*3,2) -- (pi/2*3,-2);
+        \draw[black, thick, domain=pi/2+0.5:pi/2*3-0.5] plot (\x,{tan(\x r)}) node[above]{$\tan(x)$};
+        \draw[black, densely dashed](-pi/2*3,2) -- (-pi/2*3,-2);
+        \filldraw[black] (0,0) node[below]{$O$};
+        \filldraw[black] (pi/2+0.5,-0.75) node{$\dfrac{\pi}{2}$};
+        \filldraw[black] (-pi/2-0.75,-0.75) node{$-\dfrac{\pi}{2}$};
+        \filldraw[black] (pi/2*3+0.5,-0.75) node{$\dfrac{3\pi}{2}$};
+        \filldraw[black] (-pi/2*3-0.75,-0.75) node{$-\dfrac{3\pi}{2}$};
+    \end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}{0.4\linewidth}
+    余切函数：
 
-余切函数：
-
-\begin{tikzpicture}[scale=0.7]
-    \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \draw[black, thick, domain=0.5:pi-0.5] plot (\x,{cot(\x r)}) node at(pi-1,2){$\cot(x)$};
-    \draw[black, densely dashed](pi,2) -- (pi,-2);
-    \draw[black, thick, domain=-0.5:-pi+0.5] plot (\x,{cot(\x r)}) node at(-1,2){$\cot(x)$};
-    \draw[black, densely dashed](-pi,2) -- (-pi,-2);
-    \filldraw[black] (0,0) node[below]{$O$};
-    \filldraw[black] (pi/2,0) node[below]{$\dfrac{\pi}{2}$};
-    \filldraw[black] (pi+0.5,-0.5) node{$\pi$};
-    \filldraw[black] (-pi/2-0.25,0) node[below]{$-\dfrac{\pi}{2}$};
-    \filldraw[black] (-pi-0.5,-0.5) node{$-\pi$};
-\end{tikzpicture}
+    \begin{tikzpicture}[scale=0.65]
+        \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \draw[black, thick, domain=0.5:pi-0.5] plot (\x,{cot(\x r)}) node at(pi-1,2){$\cot(x)$};
+        \draw[black, densely dashed](pi,2) -- (pi,-2);
+        \draw[black, thick, domain=-0.5:-pi+0.5] plot (\x,{cot(\x r)}) node at(-1,2){$\cot(x)$};
+        \draw[black, densely dashed](-pi,2) -- (-pi,-2);
+        \filldraw[black] (0,0) node[below]{$O$};
+        \filldraw[black] (pi/2,0) node[below]{$\dfrac{\pi}{2}$};
+        \filldraw[black] (pi+0.5,-0.5) node{$\pi$};
+        \filldraw[black] (-pi/2-0.25,0) node[below]{$-\dfrac{\pi}{2}$};
+        \filldraw[black] (-pi-0.5,-0.5) node{$-\pi$};
+    \end{tikzpicture}
+\end{minipage}
 
 切函数有如下特征：
 
@@ -221,9 +246,7 @@ $y=log_ax(a>0,a\neq 1)$为$y=a^x$的反函数：
     \item 周期性：最小正周期为$\pi$。
 \end{enumerate}
 
-$$
-    \sec x=\dfrac{1}{\cos x},\csc x=\dfrac{1}{\sin x}
-$$
+$\sec x=\dfrac{1}{\cos x},\csc x=\dfrac{1}{\sin x}$：
 
 正割函数：
 
@@ -284,31 +307,35 @@ $$
 
 \subparagraph{反三角函数} \leavevmode \medskip
 
-反正弦函数：
+\begin{minipage}{0.45\linewidth}
+    反正弦函数：
 
-\begin{tikzpicture}
-    \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \draw[black, thick, domain=-1:1] plot (\x,{rad(asin(\x))}) node[right]{$\arcsin(x)$};
-    \draw[black, densely dashed](1,pi/2) -- (0,pi/2) node[left]{$\dfrac{\pi}{2}$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, densely dashed](1,pi/2) -- (1,0) node[below]{$1$};
-    \draw[black, densely dashed](-1,-pi/2) -- (0,-pi/2) node[right]{$-\dfrac{\pi}{2}$};
-    \draw[black, densely dashed](-1,-pi/2) -- (-1,0) node[above]{$-1$};
-\end{tikzpicture}
+    \begin{tikzpicture}
+        \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \draw[black, thick, domain=-1:1] plot (\x,{rad(asin(\x))}) node[right]{$\arcsin(x)$};
+        \draw[black, densely dashed](1,pi/2) -- (0,pi/2) node[left]{$\dfrac{\pi}{2}$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, densely dashed](1,pi/2) -- (1,0) node[below]{$1$};
+        \draw[black, densely dashed](-1,-pi/2) -- (0,-pi/2) node[right]{$-\dfrac{\pi}{2}$};
+        \draw[black, densely dashed](-1,-pi/2) -- (-1,0) node[above]{$-1$};
+    \end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    反余弦函数：
 
-反余弦函数：
-
-\begin{tikzpicture}
-    \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
-    \draw[black, thick, domain=-1:1] plot (\x,{rad(acos(\x)}) node at (-2, pi){$\arccos(x)$};
-    \filldraw[black] (0,pi/2+0.5) node[right]{$\dfrac{\pi}{2}$};
-    \draw[black](1,0) -- (1,0) node[below]{$1$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, densely dashed](-1,pi) -- (0,pi) node[right]{$\pi$};
-    \draw[black, densely dashed](-1,pi) -- (-1,0) node[below]{$-1$};
-\end{tikzpicture}
+    \begin{tikzpicture}
+        \draw[-latex](-1.5,0) -- (1.5,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
+        \draw[black, thick, domain=-1:1] plot (\x,{rad(acos(\x)}) node at (-2, pi){$\arccos(x)$};
+        \filldraw[black] (0,pi/2+0.5) node[right]{$\dfrac{\pi}{2}$};
+        \draw[black](1,0) -- (1,0) node[below]{$1$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, densely dashed](-1,pi) -- (0,pi) node[right]{$\pi$};
+        \draw[black, densely dashed](-1,pi) -- (-1,0) node[below]{$-1$};
+    \end{tikzpicture}
+\end{minipage}
 
 反弦函数有如下特征：
 
@@ -327,29 +354,33 @@ $$
 
 又$f(0)=\dfrac{\pi}{2}$，所以该函数等于$\dfrac{\pi}{2}$。
 
-反正切函数：
+\begin{minipage}{0.45\linewidth}
+    反正切函数：
 
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-3,0) -- (3,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \draw[black, thick, domain=-3:3] plot (\x,{rad(atan(\x))}) node[right]{$\arctan(x)$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, densely dashed](-3,pi/2) -- (3,pi/2);
-    \draw[black, densely dashed](-3,-pi/2) -- (3,-pi/2);
-    \filldraw[black] (0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$};
-    \filldraw[black] (0.5,-pi/2-0.5) node{$-\dfrac{\pi}{2}$};
-\end{tikzpicture}
+    \begin{tikzpicture}[scale=0.75]
+        \draw[-latex](-3,0) -- (3,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \draw[black, thick, domain=-3:3] plot (\x,{rad(atan(\x))}) node[right]{$\arctan(x)$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, densely dashed](-3,pi/2) -- (3,pi/2);
+        \draw[black, densely dashed](-3,-pi/2) -- (3,-pi/2);
+        \filldraw[black] (0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$};
+        \filldraw[black] (0.5,-pi/2-0.5) node{$-\dfrac{\pi}{2}$};
+    \end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    反余切函数：
 
-反余切函数：
-
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-3,0) -- (3,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
-    \draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\textrm{arccot}(x)$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, densely dashed](-3,pi) -- (3,pi);
-    \filldraw[black] (-0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$};
-\end{tikzpicture}
+    \begin{tikzpicture}[scale=0.75]
+        \draw[-latex](-3,0) -- (3,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
+        \draw[black, thick, domain=-3:3] plot (\x,{pi/2-rad(atan(\x))}) node[right]{$\textrm{arccot}(x)$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, densely dashed](-3,pi) -- (3,pi);
+        \filldraw[black] (-0.5,pi/2-0.5) node{$\dfrac{\pi}{2}$};
+    \end{tikzpicture}
+\end{minipage}
 
 反切函数有如下特征：
 
@@ -388,27 +419,32 @@ x的不同范围对应不同的法则，经典形式如下：
 
 \subparagraph{绝对值函数} \leavevmode \medskip
 
-$
+\begin{minipage}{0.45\linewidth}
+    $
     y=\vert x\vert=\left\{
     \begin{array}{lcl}
         x,  &  & x\geqslant 0 \\
         -x, &  & x<0
     \end{array}
     \right.
-$
-
-\begin{tikzpicture}
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,2.5) node[above]{$y$};
-    \draw[black, thick, domain=0:2] plot (\x,\x);
-    \draw[black, thick, domain=-2:0] plot (\x,-\x);
-    \filldraw[black] (0.5,1.5) node{$\vert x\vert$};
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
+    $
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,2.5) node[above]{$y$};
+        \draw[black, thick, domain=0:2] plot (\x,\x);
+        \draw[black, thick, domain=-2:0] plot (\x,-\x);
+        \filldraw[black] (0.5,1.5) node{$\vert x\vert$};
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{符号函数} \leavevmode \medskip
 
-$
+\begin{minipage}{0.45\linewidth}
+    $
     y=\textrm{sgn}\,x=\left\{
     \begin{array}{lcl}
         1,  &  & x>0 \\
@@ -416,228 +452,280 @@ $
         -1, &  & x<0
     \end{array}
     \right.
-$
-
-\begin{tikzpicture}
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
-    \draw[black, thick, domain=0:2] plot (\x,1);
-    \draw[black, thick, domain=-2:0] plot (\x,-1);
-    \filldraw[black] (-1.5,1) node{$\textrm{sgn}\,x$};
-    \filldraw[black] circle (2pt) (0,0) node[below]{$O$};
-    \filldraw[white, draw=black, line width=1pt] (0,1) circle (2pt);
-    \filldraw[black] (0,1) node[left]{$1$};
-    \filldraw[white, draw=black, line width=1pt] (0,-1) circle (2pt);
-    \filldraw[black] (0,-1) node[right]{$-1$};
-\end{tikzpicture}
+    $
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
+        \draw[black, thick, domain=0:2] plot (\x,1);
+        \draw[black, thick, domain=-2:0] plot (\x,-1);
+        \filldraw[black] (-1.5,1) node{$\textrm{sgn}\,x$};
+        \filldraw[black] circle (2pt) (0,0) node[below]{$O$};
+        \filldraw[white, draw=black, line width=1pt] (0,1) circle (2pt);
+        \filldraw[black] (0,1) node[left]{$1$};
+        \filldraw[white, draw=black, line width=1pt] (0,-1) circle (2pt);
+        \filldraw[black] (0,-1) node[right]{$-1$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{取整函数} \leavevmode \medskip
 
 $x$为实数，不超过$x$的最大整数称为其整数部分$[x]$，其定义域为$R$，值域为$Z$。
 
-\begin{enumerate}
-    \item $x-1<[x]\leqslant x$。
-    \item $\lim\limits_{x\to 0^+}[x]=0$。
-    \item $\lim\limits_{x\to 0^-}[x]=-1$。
-\end{enumerate}
-
-\begin{tikzpicture}[scale=0.6]
-    \draw[-latex](-3.5,0) -- (4.5,0) node[below]{$x$};
-    \draw[-latex](0,-3.5) -- (0,3.5) node[above]{$y$};
-    \draw[black, thick, domain=1:2] plot (\x,1);
-    \draw[black, thick, domain=2:3] plot (\x,2);
-    \draw[black, thick, domain=3:4] plot (\x,3);
-    \draw[black, thick, domain=-1:0] plot (\x,-1);
-    \draw[black, thick, domain=-2:-1] plot (\x,-2);
-    \draw[black, thick, domain=-3:-2] plot (\x,-3);
-    \filldraw[black] (-2,2) node{$[x]$};
-    \filldraw[black] circle (2pt) (0,0) node[below]{$O$};
-    \foreach \x in {-2,...,4}
-    \filldraw[white, draw=black, line width=1pt] (\x,\x-1) circle (2pt);
-    \foreach \x in {3,...,-3}
-    \filldraw[black] (\x,\x) circle (2pt);
-    \foreach \x/\xtext in {-3,...,-1}
-    \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt);
-    \foreach \x/\xtext in {1,...,4}
-    \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt);
-    \foreach \x/\xtext in {1,...,3}
-    \filldraw[black] (0,\x) node[left]{\xtext} -- +(3pt, 0);
-    \foreach \x/\xtext in {-3,...,-1}
-    \filldraw[black] (0,\x) node[right]{\xtext} -- +(3pt, 0);
-\end{tikzpicture}
+\begin{minipage}{0.45\linewidth}
+    \begin{enumerate}
+        \item $x-1<[x]\leqslant x$。
+        \item $\lim\limits_{x\to 0^+}[x]=0$。
+        \item $\lim\limits_{x\to 0^-}[x]=-1$。
+    \end{enumerate}
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    \begin{tikzpicture}[scale=0.6]
+        \draw[-latex](-3.5,0) -- (4.5,0) node[below]{$x$};
+        \draw[-latex](0,-3.5) -- (0,3.5) node[above]{$y$};
+        \draw[black, thick, domain=1:2] plot (\x,1);
+        \draw[black, thick, domain=2:3] plot (\x,2);
+        \draw[black, thick, domain=3:4] plot (\x,3);
+        \draw[black, thick, domain=-1:0] plot (\x,-1);
+        \draw[black, thick, domain=-2:-1] plot (\x,-2);
+        \draw[black, thick, domain=-3:-2] plot (\x,-3);
+        \filldraw[black] (-2,2) node{$[x]$};
+        \filldraw[black] circle (2pt) (0,0) node[below]{$O$};
+        \foreach \x in {-2,...,4}
+        \filldraw[white, draw=black, line width=1pt] (\x,\x-1) circle (2pt);
+        \foreach \x in {3,...,-3}
+        \filldraw[black] (\x,\x) circle (2pt);
+        \foreach \x/\xtext in {-3,...,-1}
+        \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt);
+        \foreach \x/\xtext in {1,...,4}
+        \filldraw[black] (\x,0) node[below]{\xtext} -- ++(0, 3pt);
+        \foreach \x/\xtext in {1,...,3}
+        \filldraw[black] (0,\x) node[left]{\xtext} -- +(3pt, 0);
+        \foreach \x/\xtext in {-3,...,-1}
+        \filldraw[black] (0,\x) node[right]{\xtext} -- +(3pt, 0);
+    \end{tikzpicture}
+\end{minipage}
 
 \subsubsection{图像变换}
 \paragraph{平移变换}
 \subparagraph{左右平移} \leavevmode \medskip
 
-$f(x)$沿$x$轴左移$x_0$个单位长度得到$f(x+x_0)$，向右移动$x_0$个单位则得到$f(x-x_0)$：
-
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1);
-    \filldraw[black] (0,1.5) node{$-x^2+1$};
-    \draw[black, thick, domain=0.5:3.5] plot (\x,{-pow((\x-2),2)+1});
-    \filldraw[black] (2.5,1.5) node{$-(x-2)^2+1$};
-    \draw[black, thick, domain=-3.5:-0.5] plot (\x,{-pow((\x+2),2)+1});
-    \filldraw[black] (-2.5,1.5) node{$-(x+2)^2+1$};
-    \filldraw[black] (1,0.5) node{$\rightarrow$};
-    \filldraw[black] (-1,0.5) node{$\leftarrow$};
-\end{tikzpicture}
+\begin{minipage}{0.35\linewidth}
+    $f(x)$沿$x$轴左移$x_0$个单位长度得到$f(x+x_0)$，向右移动$x_0$个单位则得到$f(x-x_0)$：
+\end{minipage}
+\hfill
+\begin{minipage}{0.55\linewidth}
+    \begin{tikzpicture}[scale=0.9]
+        \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1);
+        \filldraw[black] (0,1.5) node{$-x^2+1$};
+        \draw[black, thick, domain=0.5:3.5] plot (\x,{-pow((\x-2),2)+1});
+        \filldraw[black] (2.5,1.5) node{$-(x-2)^2+1$};
+        \draw[black, thick, domain=-3.5:-0.5] plot (\x,{-pow((\x+2),2)+1});
+        \filldraw[black] (-2.5,1.5) node{$-(x+2)^2+1$};
+        \filldraw[black] (1,0.5) node{$\rightarrow$};
+        \filldraw[black] (-1,0.5) node{$\leftarrow$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{上下平移} \leavevmode \medskip
 
-$f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$，向下移动$y_0$个单位则得到$f(x)-y_0$：
-
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-4) -- (0,4) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1);
-    \filldraw[black] (0,-0.75) node{$-x^2+1$};
-    \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+3});
-    \filldraw[black] (0,1.5) node{$-x^2+3$};
-    \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+-1});
-    \filldraw[black] (0,-2.5) node{$-x^2-1$};
-    \filldraw[black] (-2,2.5) node{$\uparrow $};
-    \filldraw[black] (-2,-2.5) node{$\downarrow $};
-\end{tikzpicture}
+\begin{minipage}{0.45\linewidth}
+    $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$，向下移动$y_0$个单位则得到$f(x)-y_0$：
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    \begin{tikzpicture}[scale=0.9]
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-4) -- (0,4) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1);
+        \filldraw[black] (0,-0.75) node{$-x^2+1$};
+        \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+3});
+        \filldraw[black] (0,1.5) node{$-x^2+3$};
+        \draw[black, thick, domain=-1.5:1.5] plot (\x,{-\x*\x+-1});
+        \filldraw[black] (0,-2.5) node{$-x^2-1$};
+        \filldraw[black] (-2,2.5) node{$\uparrow $};
+        \filldraw[black] (-2,-2.5) node{$\downarrow $};
+    \end{tikzpicture}
+\end{minipage}
 
 \paragraph{对称变换}
 \subparagraph{上下对称} \leavevmode \medskip
 
-将$f(x)$关于$x$轴对称得到$-f(x)$：
-
-\begin{tikzpicture}
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1);
-    \filldraw[black] (0,1.5) node{$-x^2+1$};
-    \draw[black, thick, domain=-1.5:1.5] plot (\x,\x*\x-1);
-    \filldraw[black] (0,-1.5) node{$x^2-1$};
-\end{tikzpicture}
+\begin{minipage}{0.5\linewidth}
+    将$f(x)$关于$x$轴对称得到$-f(x)$：
+\end{minipage}
+\hfill
+\begin{minipage}{0.4\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=-1.5:1.5] plot (\x,-\x*\x+1);
+        \filldraw[black] (0,1.5) node{$-x^2+1$};
+        \draw[black, thick, domain=-1.5:1.5] plot (\x,\x*\x-1);
+        \filldraw[black] (0,-1.5) node{$x^2-1$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{左右对称} \leavevmode \medskip
 
-将$f(x)$关于$y$轴对称得到$f(-x)$：
-
-\begin{tikzpicture}[scale=0.8]
-    \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)});
-    \filldraw[black] (1.5,1.5) node{$\ln x$};
-    \draw[black, thick, domain=-0.25:-3.5] plot (\x,{ln(-\x)});
-    \filldraw[black] (-1.5,1.5) node{$\ln -x$};
-\end{tikzpicture}
+\begin{minipage}{0.4\linewidth}
+    将$f(x)$关于$y$轴对称得到\\$f(-x)$：
+\end{minipage}
+\hfill
+\begin{minipage}{0.5\linewidth}
+    \begin{tikzpicture}[scale=0.8]
+        \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)});
+        \filldraw[black] (1.5,1.5) node{$\ln x$};
+        \draw[black, thick, domain=-0.25:-3.5] plot (\x,{ln(-\x)});
+        \filldraw[black] (-1.5,1.5) node{$\ln -x$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{原点对称} \leavevmode \medskip
 
-将$f(x)$关于$x$轴$y$轴即关于原点对称得到$-f(-x)$：
-
-\begin{tikzpicture}[scale=0.8]
-    \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)});
-    \filldraw[black] (1.5,1.5) node{$\ln x$};
-    \draw[black, thick, domain=-0.25:-3.5] plot (\x,{-ln(-\x)});
-    \filldraw[black] (-1.5,-1.5) node{$-\ln -x$};
-\end{tikzpicture}
+\begin{minipage}{0.4\linewidth}
+    将$f(x)$关于$x$轴$y$轴即关于原点对称得到$-f(-x)$：
+\end{minipage}
+\hfill
+\begin{minipage}{0.5\linewidth}
+    \begin{tikzpicture}[scale=0.8]
+        \draw[-latex](-4,0) -- (4,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=0.25:3.5] plot (\x,{ln(\x)});
+        \filldraw[black] (1.5,1.5) node{$\ln x$};
+        \draw[black, thick, domain=-0.25:-3.5] plot (\x,{-ln(-\x)});
+        \filldraw[black] (-1.5,-1.5) node{$-\ln -x$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{反函数对称} \leavevmode \medskip
 
-将$f(x)$关于$y=x$轴对称得到$f^{-1}(x)$：
-
-\begin{tikzpicture}[scale=0.8]
-    \draw[-latex](-2,0) -- (e,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,e) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=0.25:e] plot (\x,{ln(\x)});
-    \filldraw[black] (1.5,-1.5) node{$\ln x$};
-    \draw[black, thick, domain=-1:1] plot (\x,{exp(\x)});
-    \filldraw[black] (-1.5,1.5) node{$e^x$};
-    \draw[black, densely dashed] (-2,-2) -- (e-0.5,e-0.5) node[above]{$y=x$};
-\end{tikzpicture}
+\begin{minipage}{0.55\linewidth}
+    将$f(x)$关于$y=x$轴对称得到$f^{-1}(x)$：
+\end{minipage}
+\hfill
+\begin{minipage}{0.35\linewidth}
+    \begin{tikzpicture}[scale=0.8]
+        \draw[-latex](-2,0) -- (e,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,e) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=0.25:e] plot (\x,{ln(\x)});
+        \filldraw[black] (1.5,-1.5) node{$\ln x$};
+        \draw[black, thick, domain=-1:1] plot (\x,{exp(\x)});
+        \filldraw[black] (-1.5,1.5) node{$e^x$};
+        \draw[black, densely dashed] (-2,-2) -- (e-0.5,e-0.5) node[above]{$y=x$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{函数绝对值} \leavevmode \medskip
 
-保留$f(x)$函数值在$[0,\infty]$的部分，并对$[-\infty,0]$部分进行上下对称：
-
-\begin{tikzpicture}
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=1:1.5] plot (\x,\x*\x-1);
-    \draw[black, thick, densely dashed, domain=-1:1] plot (\x,\x*\x-1);
-    \draw[black, thick, domain=-1:1] plot (\x,-\x*\x+1);
-    \draw[black, thick, domain=-1.5:-1] plot (\x,\x*\x-1);
-    \filldraw[black] (0,1.5) node{$\vert x^2-1\vert$};
-\end{tikzpicture}
+\begin{minipage}{0.5\linewidth}
+    保留$f(x)$函数值在$[0,\infty]$的部分，并对$[-\infty,0]$部分进行上下对称：
+\end{minipage}
+\hfill
+\begin{minipage}{0.4\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=1:1.5] plot (\x,\x*\x-1);
+        \draw[black, thick, densely dashed, domain=-1:1] plot (\x,\x*\x-1);
+        \draw[black, thick, domain=-1:1] plot (\x,-\x*\x+1);
+        \draw[black, thick, domain=-1.5:-1] plot (\x,\x*\x-1);
+        \filldraw[black] (0,1.5) node{$\vert x^2-1\vert$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{自变量绝对值} \leavevmode \medskip
 
-先只保留$f(x)$定义域在$[0,\infty]$的部分，然后在$[-\infty,0]$部分使用$[0,\infty]$的部分进行左右对称：
-
-\begin{tikzpicture}
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-1) -- (0,3) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=0:1.25] plot (\x,{-pow(\x,3)+1});
-    \draw[black, thick, densely dashed, domain=-1.25:0] plot (\x,{-pow(\x,3)+1});
-    \draw[black, thick, domain=-1.25:0] plot (\x,{-pow(-\x,3)+1});
-    \filldraw[black] (1,2) node{$-\vert x\vert^3+1$};
-\end{tikzpicture}
+\begin{minipage}{0.5\linewidth}
+    先只保留$f(x)$定义域在$[0,\infty]$的部分，然后在$[-\infty,0]$部分使用$[0,\infty]$的部分进行左右对称：
+\end{minipage}
+\hfill
+\begin{minipage}{0.4\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-1) -- (0,3) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=0:1.25] plot (\x,{-pow(\x,3)+1});
+        \draw[black, thick, densely dashed, domain=-1.25:0] plot (\x,{-pow(\x,3)+1});
+        \draw[black, thick, domain=-1.25:0] plot (\x,{-pow(-\x,3)+1});
+        \filldraw[black] (1,2) node{$-\vert x\vert^3+1$};
+    \end{tikzpicture}
+\end{minipage}
 
 \paragraph{伸缩变换}
 \subparagraph{水平伸缩} \leavevmode \medskip
 
-纵坐标不变，当$k>1$时，$y=f(kx)$是$y=f(x)$缩短k倍得到，当$0<k<1$时，$y=f(kx)$是$y=f(x)$伸长k倍得到：
-
-\begin{tikzpicture}[scale=0.8]
-    \draw[-latex](-5,0) -- (5,0) node[below]{$x$};
-    \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
-    \draw[black, thick, smooth, domain=-5:5] plot (\x,{sin(\x r)}) node[right]{$\sin(x)$};
-    \draw[blue, thick, smooth, domain=-5:5] plot (\x,{sin(\x/2 r)}) node[right]{$\sin(\dfrac{x}{2})$};
-    \draw[brown, thick, smooth, domain=-5:5] plot (\x,{sin(\x*2 r)}) node[right]{$\sin(2x)$};
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
-
+\begin{minipage}{0.35\linewidth}
+    纵坐标不变，当$k>1$时，$y=f(kx)$是$y=f(x)$缩短k倍得到，当$0<k<1$时，$y=f(kx)$是$y=f(x)$伸长k倍得到：
+\end{minipage}
+\hfill
+\begin{minipage}{0.55\linewidth}
+    \begin{tikzpicture}[scale=0.625]
+        \draw[-latex](-5,0) -- (5,0) node[below]{$x$};
+        \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
+        \draw[black, thick, smooth, domain=-5:5] plot (\x,{sin(\x r)}) node[right]{$\sin(x)$};
+        \draw[blue, thick, smooth, domain=-5:5] plot (\x,{sin(\x/2 r)}) node[right]{$\sin(\dfrac{x}{2})$};
+        \draw[brown, thick, smooth, domain=-5:5] plot (\x,{sin(\x*2 r)}) node[right]{$\sin(2x)$};
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subparagraph{垂直伸缩} \leavevmode \medskip
 
-横坐标不变，$y=kf(x)$的对应纵坐标为$y=f(x)$对应纵坐标的$k$倍。
-
-\begin{tikzpicture}[scale=0.8]
-    \draw[-latex](-5,0) -- (5,0) node[below]{$x$};
-    \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
-    \draw[black, thick, smooth, domain=-5:5] plot (\x,{sin(\x r)}) node[right]{$\sin(x)$};
-    \draw[blue, thick, smooth, domain=-5:5] plot (\x,{sin(\x r)/2}) node at(5.5,1){$\dfrac{sin(x)}{2}$};
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
+\begin{minipage}{0.35\linewidth}
+    横坐标不变，$y=kf(x)$的对应纵坐标为$y=f(x)$对应纵坐标的$k$倍。
+\end{minipage}
+\hfill
+\begin{minipage}{0.55\linewidth}
+    \begin{tikzpicture}[scale=0.65]
+        \draw[-latex](-5,0) -- (5,0) node[below]{$x$};
+        \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
+        \draw[black, thick, smooth, domain=-5:5] plot (\x,{sin(\x r)}) node[right]{$\sin(x)$};
+        \draw[blue, thick, smooth, domain=-5:5] plot (\x,{sin(\x r)/2}) node at(5.5,1){$\dfrac{sin(x)}{2}$};
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subsection{极坐标系图像}
 \subsubsection{描点法}
 \paragraph{心形线（外摆线）} \leavevmode \medskip
 
-心形线又称为心脏线，表示是一个圆上的固定一点在它绕着与其相切且半径相同的另外一个圆周滚动时所形成的轨迹。
+\begin{minipage}{0.55\linewidth}
+    心形线又称为心脏线，表示是一个圆上的固定一点在它绕着与其相切且半径相同的另外一个圆周滚动时所形成的轨迹。
 
-表达式：水平为$r=a(1\pm\cos\theta)$，垂直为$r=a(1\pm\sin\theta)$。一般为$r=a(1-\cos\theta)$即为下图所示，如果里面的符号为＋则心尖开口向左。
+    表达式：水平为$r=a(1\pm\cos\theta)$，垂直为$r=a(1\pm\sin\theta)$。一般为$r=a(1-\cos\theta)$即为下图所示，如果里面的符号为＋则心尖开口向左。
 
-其中$r$为线的极径，$\theta$为极角，$a$为形状参数且$a>0$，周期为$2\pi$。
+    其中$r$为线的极径，$\theta$为极角，$a$为形状参数且$a>0$，周期为$2\pi$。
+\end{minipage}
+\hfill
+\begin{minipage}{0.35\linewidth}
+    \begin{tikzpicture}[scale=0.8]
+        \draw[-latex](-5,0) -- (1,0) node[below]{$x$};
+        \draw[-latex](0,-3) -- (0,3) node[above]{$y$};
+        \draw[black, thick, domain=0:360,smooth,variable=\t, samples=300] plot ({\t}:{2*(1-cos(\t))});
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw (-1,0) circle [radius=1];
+        \draw (-3,0) circle [radius=1];
+    \end{tikzpicture}
+\end{minipage}
 
 在直角坐标系下表达式：$x^2+y^2+a\cdot x=a\cdot\sqrt{x^2+y^2}$和$x^2+y^2-a\cdot x=a\cdot\sqrt{x^2+y^2}$。
 
-参数方程：$x=a\cdot(2\cdot\cos(t)-cos(2\cdot t))$与$y=a\cdot(2\cdot\sin(t)-sin(2\cdot t))$
-
-\begin{tikzpicture}[scale=0.8]
-    \draw[-latex](-5,0) -- (1,0) node[below]{$x$};
-    \draw[-latex](0,-3) -- (0,3) node[above]{$y$};
-    \draw[black, thick, domain=0:360,smooth,variable=\t, samples=300] plot ({\t}:{2*(1-cos(\t))});
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
+参数方程：$x=a\cdot(2\cdot\cos(t)-cos(2\cdot t))$与$y=a\cdot(2\cdot\sin(t)-sin(2\cdot t))$。
 
 水平心形线对应参数： \leavevmode \medskip
 
@@ -650,18 +738,22 @@ $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$，向下移动$y_0$个
 
 \paragraph{玫瑰线} \leavevmode \medskip
 
-表达式：$r=a\sin(n\theta)$，周期为$\dfrac{2\pi}{n}$。
+\begin{minipage}{0.55\linewidth}
+    表达式：$r=a\sin(n\theta)$，周期为$\dfrac{2\pi}{n}$。
 
-当$n$为3时为三叶，2时为四叶，$\dfrac{3}{2}$为六叶。三叶时周期为$\dfrac{2\pi}{3}$。
+    当$n$为3时为三叶，2时为四叶，$\dfrac{3}{2}$为六叶。三叶时周期为$\dfrac{2\pi}{3}$。
 
-直角坐标系下表达式：$x=a\cdot\sin(n\cdot\theta)\cdot\cos(\theta)$与$y=a\cdot\sin(n\cdot)\cdot\sin(\theta)$
-
-\begin{tikzpicture}[scale=0.8]
-    \draw[-latex](-3,0) -- (3,0) node[below]{$x$};
-    \draw[-latex](0,-pi) -- (0,pi/2) node[above]{$y$};
-    \draw[domain=0:180,samples=100] plot (\x:{3*sin(\x*3)});
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
+    直角坐标系下表达式：$x=a\cdot\sin(n\cdot\theta)\cdot\cos(\theta)$与$y=a\cdot\sin(n\cdot)\cdot\sin(\theta)$
+\end{minipage}
+\hfill
+\begin{minipage}{0.35\linewidth}
+    \begin{tikzpicture}[scale=0.8]
+        \draw[-latex](-3,0) -- (3,0) node[below]{$x$};
+        \draw[-latex](0,-pi) -- (0,pi/2) node[above]{$y$};
+        \draw[domain=0:180,samples=100] plot (\x:{3*sin(\x*3)});
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
 
 三叶玫瑰线对应参数： \leavevmode \medskip
 
@@ -674,14 +766,18 @@ $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$，向下移动$y_0$个
 
 \paragraph{阿基米德螺线} \leavevmode \medskip
 
-表达式：$r=a\theta$，其中$a>0$，$\theta\geqslant 0$由0开始增大时$r$也在不断增大。
-
-\begin{tikzpicture}[scale=0.2]
-    \draw[-latex](-10,0) -- (15,0) node[below]{$x$};
-    \draw[-latex](0,-15) -- (0,10) node[above]{$y$};
-    \draw[domain=0:720,samples=100] plot (\x:{rad(\x)});
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
+\begin{minipage}{0.5\linewidth}
+    表达式：$r=a\theta$，其中$a>0$，$\theta\geqslant 0$由0开始增大时$r$也在不断增大。
+\end{minipage}
+\hfill
+\begin{minipage}{0.4\linewidth}
+    \begin{tikzpicture}[scale=0.2]
+        \draw[-latex](-10,0) -- (15,0) node[below]{$x$};
+        \draw[-latex](0,-15) -- (0,10) node[above]{$y$};
+        \draw[domain=0:720,samples=100] plot (\x:{rad(\x)});
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
 
 \paragraph{伯努利双扭线} \leavevmode \medskip
 
@@ -711,16 +807,20 @@ $f(x)$沿$y$轴上移$y_0$个单位长度得到$f(x)+y_0$，向下移动$y_0$个
 
 \subsubsection{直角坐标系下画极坐标图像}
 
-令$\theta$为$x$，令$r$为$y$。如心形线$r=2(1-\cos\theta)$：
+\begin{minipage}{0.5\linewidth}
+    令$\theta$为$x$，令$r$为$y$。如心形线$r=2(1-\cos\theta)$：
 
-\begin{tikzpicture}[scale=0.5]
-    \draw[-latex](-5,0) -- (5,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,5) node[above]{$y$};
-    \draw[black, thick, smooth, domain=-5:5] plot (\x,{2*(1-cos(\x r))}) node at (0,4){$2(1-\cos(\theta))$};
-    \filldraw[black] (0,0) node[below]{$O$};
-\end{tikzpicture}
-
-按直角坐标系的图就可以计算出对应的$r$从而能画出对应的图像。
+    按直角坐标系的图就可以计算出对应的$r$从而能画出对应的图像。
+\end{minipage}
+\hfill
+\begin{minipage}{0.4\linewidth}
+    \begin{tikzpicture}[scale=0.5]
+        \draw[-latex](-5,0) -- (5,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,5) node[above]{$y$};
+        \draw[black, thick, smooth, domain=-5:5] plot (\x,{2*(1-cos(\x r))}) node at (0,4){$2(1-\cos(\theta))$};
+        \filldraw[black] (0,0) node[below]{$O$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subsection{参数法}
 
@@ -737,33 +837,56 @@ $
 
 摆线，又称旋轮线、圆滚线，是一个圆沿一条直线滚动时，圆边界上一定点所形成的轨迹。
 
-令圆半径为$r$，摆点与圆心所成直线所转动夹角对应弧度为$t$，其中$t\in[0,2\pi]$，所对应参数方程为：
-
-$$
+令圆半径为$r$，摆点与圆心所成直线所转动夹角对应弧度为$t$，其中$t\in[0,2\pi]$，所对应参数方程为：$
     \left\{
     \begin{array}{lcl}
         x=r(t-\sin t) \\
         y=r(1-\cos t)
     \end{array}
     \right.
-$$
+$
+
+\begin{tikzpicture}[scale=1.5]
+    \draw[-latex](-1.5,0) -- (5,0) node[below]{$x$};
+    \draw[-latex](0,-0.5) -- (0,2) node[above]{$y$};
+    \filldraw[black] (0,0) node[below]{$O$};
+    \draw[black,scale=0.35, domain=-1.7:2*4.0, smooth, variable=\t ]
+    plot ( {2*(\t-sin(\t r))}, {2*(1-cos(\t r))});
+    \draw (0.7,0.7) circle [radius=0.7];
+    \draw[black](0.7,0) -- (0.7,1.4);
+    \draw[black, densely dashed](2.25,0) -- (2.25,1.4);
+    \filldraw[black] (2.5,0.625) node{$2a$};
+\end{tikzpicture}
 
 \subsubsection{星形线（内摆线）}
 
-与半径为$r$的定圆内切的半径为$\dfrac{r}{4}$的动圆沿定圆无滑动地滚动，动圆上一点的轨迹称为星形线。
+\begin{minipage}{0.5\linewidth}
+    与半径为$r$的定圆内切的半径为$\dfrac{r}{4}$的动圆沿定圆无滑动地滚动，动圆上一点的轨迹称为星形线。
 
-令$t$表示摆点与圆心的连线所构成夹角的弧度，其中$t\in[0,2\pi]$，得对应参数方程：
-
-$$
+    令$t$表示摆点与圆心的连线所构成夹角的弧度，其中$t\in[0,2\pi]$，得对应参数方程：$
     \left\{
     \begin{array}{lcl}
         x=r\cos^3t \\
         y=r\sin^3t
     \end{array}
     \right.
-$$
+    $
 
-由$\cos^2t+\sin^2t=1$得到直角坐标方程：$x^{\frac{2}{3}}+y^{\frac{2}{3}}=r^{\frac{2}{3}}$
+    由$\cos^2t+\sin^2t=1$得到直角坐标方程：$x^{\frac{2}{3}}+y^{\frac{2}{3}}=r^{\frac{2}{3}}$。
+\end{minipage}
+\hfill
+\begin{minipage}{0.4\linewidth}
+    \begin{tikzpicture}[scale=2]
+        \draw[-latex](-1.25,0) -- (1.25,0) node[below]{$x$};
+        \draw[-latex](0,-1.25) -- (0,1.25) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black,scale=1, domain=-1.7:2*4.0, smooth, variable=\t ]
+        plot ( {(pow(cos(\t r),3))}, {(pow(sin(\t r),3))});
+        \filldraw[black] (-0.25,0.125) node{$a$};
+        \draw (0,0) circle [radius=1];
+        \draw (0.75,0) circle [radius=0.25];
+    \end{tikzpicture}
+\end{minipage}
 
 \section{常用基础知识}
 \subsection{数列}
diff --git a/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf b/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf
index f5e1a55..b60af38 100644
Binary files a/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf and b/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf differ
diff --git a/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex b/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex
index e0a0060..d43f1e6 100644
--- a/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex
+++ b/advanced-math/knowledge/1-function-and-limit/function-and-limit.tex
@@ -830,24 +830,30 @@ $
 
 \subsection{\texorpdfstring{$\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$}{}}
 
-证明：当$x\to 0$时$x\in[0,\dfrac{\pi}{2}]$。
+证明：
 
-\begin{tikzpicture}[scale=1.5]
-    \draw (0,0) circle (1);
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black](0,0) -- (1,0) node[right]{$A$};
-    \draw[black](0,0) -- (1/2,{sqrt(3)/2}) node[above]{$B$};
-    \draw[black](1/2,{sqrt(3)/2}) -- (1/2,0) node[below]{$D$};
-    \draw[black](1,0) -- (1,{sqrt(3)}) node[above]{$C$};
-    \draw[black](1,0) -- (1/2,{sqrt(3)/2});
-    \draw[black](1,{sqrt(3)}) -- (1/2,{sqrt(3)/2});
-\end{tikzpicture}
+\begin{minipage}{0.7\linewidth}
+    当$x\to 0$时$x\in[0,\dfrac{\pi}{2}]$。
 
-设$\angle AOB$的弧度为$x$，圆$O$的半径为$1$，则$OD=\sin x$。
+    设$\angle AOB$的弧度为$x$，圆$O$的半径为$1$，则$OD=\sin x$。
 
-则$S_\vartriangle AOB=\dfrac{\sin x}{2}$。根据扇形面积公式：$S_{\text{扇形}}AOB=\dfrac{x}{2}$。
+    则$S_\vartriangle AOB=\dfrac{\sin x}{2}$。根据扇形面积公式：$S_{\text{扇形}}AOB=\dfrac{x}{2}$。
 
-又$\because CA=\tan x$，则$S_\vartriangle AOC=\dfrac{\tan x}{2}$。
+    又$\because CA=\tan x$，则$S_\vartriangle AOC=\dfrac{\tan x}{2}$。
+\end{minipage}
+\hfill
+\begin{minipage}{0.2\linewidth}
+    \begin{tikzpicture}[scale=1.125]
+        \draw (0,0) circle (1);
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black](0,0) -- (1,0) node[right]{$A$};
+        \draw[black](0,0) -- (1/2,{sqrt(3)/2}) node[above]{$B$};
+        \draw[black](1/2,{sqrt(3)/2}) -- (1/2,0) node[below]{$D$};
+        \draw[black](1,0) -- (1,{sqrt(3)}) node[above]{$C$};
+        \draw[black](1,0) -- (1/2,{sqrt(3)/2});
+        \draw[black](1,{sqrt(3)}) -- (1/2,{sqrt(3)/2});
+    \end{tikzpicture}
+\end{minipage}
 
 根据图，在$x\in[0,\dfrac{\pi}{2}]$，$\sin x<x<\tan x$。
 
@@ -957,64 +963,80 @@ $=e^{\lim\limits_{x\to\infty}\frac{\left(\frac{1}{1+\frac{1}{x}}\right)\cdot\lef
 
 \subsubsection{可去间断点（可补间断点）}
 
-\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=A\neq f(x_0)$（甚至可以没有定义）。
-
-\begin{tikzpicture}
-    \draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=-0.5:2] plot (\x,{pow(e,\x-1)});
-    \filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
-    \draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
-    \draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
-    \draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
-    \filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
-    \filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
-\end{tikzpicture}
+\begin{minipage}{0.6\linewidth}
+    \textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=A\neq f(x_0)$（甚至可以没有定义）。
+\end{minipage}
+\hfill
+\begin{minipage}{0.3\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=-0.5:2] plot (\x,{pow(e,\x-1)});
+        \filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
+        \draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
+        \draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
+        \draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
+        \filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
+        \filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
+    \end{tikzpicture}
+\end{minipage}
 
 \subsubsection{跳跃间断点}
 
-\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0^-}f(x)$与$\lim\limits_{x\to x_0^+}f(x)$都存在，但是$\lim\limits_{x\to x_0^-}f(x)\neq\lim\limits_{x\to x_0^+}f(x)$。
-
-\begin{tikzpicture}
-    \draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=-0.5:1] plot (\x,{pow(e,\x-1)});
-    \draw[black, thick, domain=1:1.5] plot (\x,{pow(e,\x-1)+1});
-    \filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
-    \draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
-    \draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
-    \draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
-    \filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
-    \filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
-\end{tikzpicture}
+\begin{minipage}{0.6\linewidth}
+    \textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0^-}f(x)$与$\lim\limits_{x\to x_0^+}f(x)$都存在，但是$\lim\limits_{x\to x_0^-}f(x)\neq\lim\limits_{x\to x_0^+}f(x)$。
+\end{minipage}
+\hfill
+\begin{minipage}{0.3\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-0.5,0) -- (3,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,3) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=-0.5:1] plot (\x,{pow(e,\x-1)});
+        \draw[black, thick, domain=1:1.5] plot (\x,{pow(e,\x-1)+1});
+        \filldraw[white, draw=black, line width=1pt] (1,1) circle (2pt);
+        \draw[black, densely dashed](1,2) -- (0,2) node[left]{$B$};
+        \draw[black, densely dashed](1,1) -- (0,1) node[left]{$A$};
+        \draw[black, densely dashed](1,2) -- (1,0) node[below]{$x_0$};
+        \filldraw[black] (1,2) circle (2pt) node[above]{$f(x_0)=B$};
+        \filldraw[black] (1,1) node[right]{$\lim\limits_{x\to x_0}=A$};
+    \end{tikzpicture}
+\end{minipage} \medskip
 
 可去间断点与跳跃间断点的左右极限都存在的间断点都称为第一类间断点。
 
 \subsubsection{无穷间断点}
 
-\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=\infty$，或至少一个方向为无穷大（定义分歧）。如$y=\dfrac{1}{x}$在$x=0$处为无穷间断点。
-
-\begin{tikzpicture}
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=0.5:2] plot (\x,{pow(\x,-1)});
-    \draw[black, thick, domain=-2:-0.5] plot (\x,{pow(\x,-1)});
-\end{tikzpicture}
+\begin{minipage}{0.6\linewidth}
+    \textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)=\infty$，或至少一个方向为无穷大（定义分歧）。如$y=\dfrac{1}{x}$在$x=0$处为无穷间断点。
+\end{minipage}
+\hfill
+\begin{minipage}{0.3\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-2) -- (0,2) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=0.5:2] plot (\x,{pow(\x,-1)});
+        \draw[black, thick, domain=-2:-0.5] plot (\x,{pow(\x,-1)});
+    \end{tikzpicture}
+\end{minipage}
 
 \subsubsection{振荡间断点}
 
-\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)$为振荡不存在。如$\lim\limits_{x\to 0}\sin\dfrac{1}{x}$的$x=0$就是振荡间断点。
-
-\begin{tikzpicture}
-    \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
-    \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick, domain=0.01:2] plot (\x,{sin(pow(\x,-1) r)});
-    \draw[black, thick, domain=-2:-0.01] plot (\x,{sin(pow(\x,-1) r)});
-\end{tikzpicture}
+\begin{minipage}{0.6\linewidth}
+    \textcolor{violet}{\textbf{定义：}}若$\lim\limits_{x\to x_0}f(x)$为振荡不存在。如$\lim\limits_{x\to 0}\sin\dfrac{1}{x}$的$x=0$就是振荡间断点。
+\end{minipage}
+\hfill
+\begin{minipage}{0.3\linewidth}
+    \begin{tikzpicture}
+        \draw[-latex](-2,0) -- (2,0) node[below]{$x$};
+        \draw[-latex](0,-1.5) -- (0,1.5) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick, domain=0.01:2] plot (\x,{sin(pow(\x,-1) r)});
+        \draw[black, thick, domain=-2:-0.01] plot (\x,{sin(pow(\x,-1) r)});
+    \end{tikzpicture}
+\end{minipage}
 
 无穷间断点与振荡间断点的左右极限都不存在的点都是第二类间断点。
 
diff --git a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf
index 767ee03..604580e 100644
Binary files a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf and b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf differ
diff --git a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex
index 38e668b..b86cda7 100644
--- a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex
+++ b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.tex
@@ -487,36 +487,40 @@ $\therefore \sqrt{2}$。
 
 有一个边长为$x$的正方形，变化了$\Delta x$，其面积$\Delta S=(x+\Delta x)^2-x^2=2x\Delta x+(\Delta x)^2$。
 
-当$\Delta x\to 0$时，将这个变化定义为$2x\cdot\Delta x+o(\Delta x)$，前项为线性主部，后面为误差。这个就是$S$的微分。
+\begin{minipage}{0.45\linewidth}
+    当$\Delta x\to 0$时，将这个变化定义为$2x\cdot\Delta x+o(\Delta x)$，前项为线性主部，后面为误差。这个就是$S$的微分。
 
-增量$\Delta y=f(x_0+\Delta)-f(x_0)=A\Delta x+o(\Delta x)$，这个$A\Delta x$定义为$\textrm{d}y$，叫做$y$的微分。
+    增量$\Delta y=f(x_0+\Delta)-f(x_0)=A\Delta x+o(\Delta x)$，这个$A\Delta x$定义为$\textrm{d}y$，叫做$y$的微分。
 
-$\therefore \textrm{d}y\vert_{x=x_0}=A\Delta x=y'(x_0)\cdot\Delta x=y'(x_0)\cdot\textrm{d}x$
+    $\therefore \textrm{d}y\vert_{x=x_0}=A\Delta x=y'(x_0)\cdot\Delta x=y'(x_0)\cdot\textrm{d}x$。
 
-由此，可导必可微，可微必可导。
-
-\begin{tikzpicture}[scale=0.9]
-    \draw[-latex](-0.5,0) -- (4.5,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
-    \draw[black, thick, domain=-0.5:3] plot (\x,{pow(\x-1,2)/2+1}) node[above]{$y(x)$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, densely dashed](1.5,1.125) -- (1.5,0) node[below]{$x_0$};
-    \draw[black, densely dashed](1.5,1.125) -- (0,1.125) node[left]{$y_0$};
-    \draw[black, densely dashed](3,3) -- (3,0) node[below]{$x_0+\Delta x$};
-    \draw[black, densely dashed](3,3) -- (0,3) node[left]{$y_0+\Delta x$};
-    \draw[black, densely dashed](3,1.875) -- (0,0.375) node[left]{$\textrm{d}y\cdot x+b$};
-    \draw[<->, black](1.5,1.125) -- (3,1.125);
-    \draw[<->, black](4,1.125) -- (4,3);
-    \draw[<->, black](3.25,1.125) -- (3.25,1.875);
-    \draw[<->, black](3.25,3) -- (3.25,1.875);
-    \draw[black](3,3) -- (4.5,3);
-    \draw[black](3,1.125) -- (4.5,1.125);
-    \draw[black](3,1.875) -- (3.75,1.875);
-    \filldraw[black] (2.25,0.75) node{$\Delta x$};
-    \filldraw[black] (4.3,2) node{$\Delta y$};
-    \filldraw[black] (3.5,1.5) node{\scriptsize{$\textrm{d}y$}};
-    \filldraw[black] (3.5,2.5) node{\scriptsize{$o(\Delta x)$}};
-\end{tikzpicture}
+    由此，可导必可微，可微必可导。
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    \begin{tikzpicture}[scale=0.9]
+        \draw[-latex](-0.5,0) -- (4.5,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0,4) node[above]{$y$};
+        \draw[black, thick, domain=-0.5:3] plot (\x,{pow(\x-1,2)/2+1}) node[above]{$y(x)$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, densely dashed](1.5,1.125) -- (1.5,0) node[below]{$x_0$};
+        \draw[black, densely dashed](1.5,1.125) -- (0,1.125) node[left]{$y_0$};
+        \draw[black, densely dashed](3,3) -- (3,0) node[below]{$x_0+\Delta x$};
+        \draw[black, densely dashed](3,3) -- (0,3) node[left]{$y_0+\Delta x$};
+        \draw[black, densely dashed](3,1.875) -- (0,0.375) node[left]{$\textrm{d}y\cdot x+b$};
+        \draw[<->, black](1.5,1.125) -- (3,1.125);
+        \draw[<->, black](4,1.125) -- (4,3);
+        \draw[<->, black](3.25,1.125) -- (3.25,1.875);
+        \draw[<->, black](3.25,3) -- (3.25,1.875);
+        \draw[black](3,3) -- (4.5,3);
+        \draw[black](3,1.125) -- (4.5,1.125);
+        \draw[black](3,1.875) -- (3.75,1.875);
+        \filldraw[black] (2.25,0.75) node{$\Delta x$};
+        \filldraw[black] (4.3,2) node{$\Delta y$};
+        \filldraw[black] (3.5,1.5) node{\scriptsize{$\textrm{d}y$}};
+        \filldraw[black] (3.5,2.5) node{\scriptsize{$o(\Delta x)$}};
+    \end{tikzpicture}
+\end{minipage} \medskip
 
 所以可微就是用简单线性取代复杂线性，如图用直线取替代曲线。微分就是瞬时改变量，而导数就是瞬时改变速率。
 
diff --git a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf
index 773bb15..36f6ead 100644
Binary files a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf and b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.pdf differ
diff --git a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex
index d6c3086..807f01d 100644
--- a/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex
+++ b/advanced-math/knowledge/3-applications-of-derivatives/applications-of-derivatives.tex
@@ -31,8 +31,6 @@
 % 圆圈序号
 \usepackage{mathtools}
 % 有字的长箭头
-\usepackage{yhmath}
-% 弧线标识
 \usetikzlibrary{decorations.pathreplacing}
 % tikz的大括号
 \usepackage{yhmath}
@@ -64,28 +62,32 @@ $\text{罗尔定理}\xrightleftharpoons[\text{特例：}f(a)=f(b)]{\text{泛化
 
 罗尔定理\textcolor{violet}{\textbf{定义：}}
 
-\begin{enumerate}
-    \item $f(x)$在$[a,b]$上连续。
-    \item $f(x)$在$(a,b)$内可导。
-    \item $f(a)=f(b)$。
-\end{enumerate}
-
-则$\exists\,\xi\in(a,b)$，使得$f'(\xi)=0$。
-
-\begin{tikzpicture}[scale=0.7]
-    \draw[-latex](-0.5,0) -- (8,0) node[below]{$x$};
-    \draw[-latex](0,-0.5) -- (0, 4) node[above]{$y$};
-    \filldraw[black] (0,0) node[below]{$O$};
-    \draw[black, thick,domain=-0.5:8] plot (\x, {sin((\x-0.5) r)+2});
-    \filldraw[black] (6,3.5) node {$y=f(x)$};
-    \draw[densely dashed](0.5,2) -- (0.5+2*pi, 2);
-    \draw[densely dashed](0.5,2) -- (0.5, 0) node[below]{$a$};
-    \draw[densely dashed](0.5+2*pi,2) -- (0.5+2*pi, 0) node[below]{$b$};
-    \draw[densely dashed](0.5+pi/2,3) -- (0.5+pi/2, 0) node[below]{$\xi_1$};
-    \draw[densely dashed](0.5+pi/2*3,1) -- (0.5+pi/2*3, 0) node[below]{$\xi_2$};
-    \draw[black](pi/2-0.5,3) -- (1.5+pi/2,3);
-    \draw[black](pi/2*3-0.5,1) -- (1.5+pi/2*3,1);
-\end{tikzpicture}
+\begin{minipage}{0.45\linewidth}
+    \begin{enumerate}
+        \item $f(x)$在$[a,b]$上连续。
+        \item $f(x)$在$(a,b)$内可导。
+        \item $f(a)=f(b)$。
+    \end{enumerate}
+    
+    则$\exists\,\xi\in(a,b)$，使得$f'(\xi)=0$。
+\end{minipage}
+\hfill
+\begin{minipage}{0.45\linewidth}
+    \begin{tikzpicture}[scale=0.7]
+        \draw[-latex](-0.5,0) -- (8,0) node[below]{$x$};
+        \draw[-latex](0,-0.5) -- (0, 4) node[above]{$y$};
+        \filldraw[black] (0,0) node[below]{$O$};
+        \draw[black, thick,domain=-0.5:8] plot (\x, {sin((\x-0.5) r)+2});
+        \filldraw[black] (6,3.5) node {$y=f(x)$};
+        \draw[densely dashed](0.5,2) -- (0.5+2*pi, 2);
+        \draw[densely dashed](0.5,2) -- (0.5, 0) node[below]{$a$};
+        \draw[densely dashed](0.5+2*pi,2) -- (0.5+2*pi, 0) node[below]{$b$};
+        \draw[densely dashed](0.5+pi/2,3) -- (0.5+pi/2, 0) node[below]{$\xi_1$};
+        \draw[densely dashed](0.5+pi/2*3,1) -- (0.5+pi/2*3, 0) node[below]{$\xi_2$};
+        \draw[black](pi/2-0.5,3) -- (1.5+pi/2,3);
+        \draw[black](pi/2*3-0.5,1) -- (1.5+pi/2*3,1);
+    \end{tikzpicture}
+\end{minipage}
 
 \subsection{拉格朗日中值定理}
 
diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf
index 462509b..a440449 100644
Binary files a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf and b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf differ
diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex
index 753b696..acf93a6 100644
--- a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex
+++ b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex
@@ -315,6 +315,10 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
 
 \subsection{性质}
 
+\textcolor{aqua}{\textbf{定理：}}定积分由积分上下限与函数关系确定，与积分变量无关。$\int_a^xf(x)\,\textrm{d}x=\int_a^xf(t)\,\textrm{d}t=\varPhi(x)$。
+
+\subsubsection{计算性质}
+
 设函数$f(x)$在区间$[a,b]$上连续，则得到定积分基本计算性质：
 
 \begin{enumerate}
@@ -329,8 +333,15 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
     \item 已知$f(x)\in[m,M]$在$[a,b]$上成立，则$m(b-a)\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant M(a-b)$。
     \item 估值定理：当$M$，$m$分别为$f(x)$在$[a,b]$上的最大值和最小值，则$m(b-a)\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant M(b-a)$。
     \item 积分中值定理：$\exists\,\xi\in[a,b]$，使得$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$。
+    \item 积分中值定理推广：设$f(x)\in[a,b]$，$\exists\,\xi\in(a,b)$，使得$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$。
 \end{enumerate}
 
+证明第八条：已知$-\vert f(x)\vert\leqslant f(x)\leqslant\vert f(x)\vert$。
+
+即得到$-\int_a^b\vert f(x)\vert\,\textrm{d}x\leqslant\int_a^bf(x)\,\textrm{d}x\leqslant\int_a^b\vert f(x)\vert\,\textrm{d}x$。
+
+从而得证。
+
 证明第十一条积分中值定理：
 
 设函数$f(x)$在区间$[a,b]$上连续，因为闭区间上连续函数必然有最大最小值，所以设最大值为$M$，最小值为$m$，$M\geqslant m$。
@@ -349,7 +360,11 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
 
 对$F(x)$使用拉格朗日中值定理：$F(b)-F(a)=F'(\xi)(b-a)$，即$\int_a^bf(x)\,\textrm{d}x=f(\xi)(b-a)$，其中$\xi\in(a,b)\subset[a,b]$。
 
-定积分的存在性性质：
+证明第十二条积分中值定理的推广。令$F(x)=\int_a^xf(t)\,\textrm{d}t$，$F'(x)=f(x)$。
+
+$\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)=F'(\xi)(b-a)=f(\xi)(b-a)$（$a<\xi<b$）。
+
+\subsubsection{存在性性质}
 
 \begin{enumerate}
     \item 设函数$f(x)$在区间$[a,b]$上连续，则$f(x)$在该区间上可积。
@@ -357,7 +372,7 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo
     \item 设函数$f(x)$在区间$[a,b]$上有界，且只有有限个间断点，则$f(x)$在该区间上可积。
 \end{enumerate}
 
-定积分与函数性质：
+\subsubsection{定积分与函数性质}
 
 \begin{enumerate}
     \item 若函数$f(x)$是周期函数且周期为$T$，$\int_a^{a+T}f(x)\,\textrm{d}x=\int_0^Tf(x)\,\textrm{d}x$对于$\forall a$成立。
@@ -436,8 +451,6 @@ $\therefore\int_0^\pi x\sin^9x\,\textrm{d}x=\dfrac{\pi}{2}\int_0^\pi\sin^9x\,\te
 
 设$f(x)$在$[a,b]$上连续，且$\Phi(x)=\int_a^xf(t)\,\textrm{d}t(x\in[a,b])$，这个函数就是积分上限函数或叫积分变限函数（如果$\int_x^af(t)\,\textrm{d}t$就是变下限积分或积分下限函数）。
 
-积分值与积分变量无关，即$\int_a^xf(x)\,\textrm{d}x=\int_a^xf(t)\,\textrm{d}t$。
-
 对变限积分$\int_{a}^xf(t)\,\textrm{d}t$求导得到$f(t)$，再求导就得到$f'(t)$。
 
 定限积分是一个数值，而变限积分是一个函数。
diff --git a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf
new file mode 100644
index 0000000..1349349
Binary files /dev/null and b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf differ
diff --git a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex
index f7eed2e..f983798 100644
--- a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex
+++ b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex
@@ -1,7 +1,10 @@
 \documentclass[UTF8, 12pt]{ctexart}
 % UTF8编码，ctexart现实中文
-\usepackage{color}
+\usepackage{xcolor}
 % 使用颜色
+\definecolor{orange}{RGB}{255,127,0} 
+\definecolor{violet}{RGB}{192,0,255} 
+\definecolor{aqua}{RGB}{0,255,255} 
 \usepackage{geometry}
 \setcounter{tocdepth}{4}
 \setcounter{secnumdepth}{4}
@@ -22,6 +25,10 @@
 % 数学公式
 \usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref}
 % 超链接
+\usepackage{pifont}
+% 圆圈序号
+\usepackage{tikz}
+% 绘图
 \author{Didnelpsun}
 \title{多元函数微分学}
 \date{}
@@ -34,6 +41,223 @@
 \newpage
 \pagestyle{plain}
 \setcounter{page}{1}
-\section{}
+\section{基本概念}
+\subsection{平面与点}
+\subsubsection{平面点集}
+
+\textcolor{violet}{\textbf{定义：}}在平面上建立直角坐标系$xOy$，则平面上的点可用两个实数组的有序数组$(x,y)$表示，而二元函数$f(x,y)$的定义域是$(x,y)$为元素的几何，所以$f(x,y)$的定义域就是\textbf{平面上的点集}。
+
+\subsubsection{距离}
+
+\textcolor{aqua}{\textbf{定理：}}平面上任意两点$M_1(x_1,y_1)$与$M_2(x_2,y_2)$之间距离定义为$\rho(M_1,M_2)$\\$=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$。
+
+$\rho(M_1,M_2)$满足：
+
+\begin{itemize}
+    \item 非负性：$\rho(M_1,M_2)\geqslant0$。
+    \item 对称性：$\rho(M_1,M_2)=\rho(M_2,M_1)$。
+    \item 三角不等式：$\rho(M_1,M_3)\leqslant\rho(M_1,M_2)+\rho(M_2,M_3)$。
+\end{itemize}
+
+\subsubsection{邻域}
+
+设$M_0$为平面上一点，$\delta>0$，则平面上以$M_0$为圆心，$\delta$为半径的圆的内部称为$M_0$的\textbf{$\delta$领域}，记为$U(M_0,\delta)$。
+
+若领域中去掉圆心$M_0$，称为$M_0$的\textbf{$\delta$去心邻域}，记为$\mathring{U}(M_0,\delta)$。
+
+\subsubsection{点的分类}
+
+\textcolor{violet}{\textbf{定义：}}设$M$为平面上一个点，若存在$\delta>0$，使得$U(M,\delta)\subset E$，则$M$为点集$E$的\textbf{内点}。
+
+\textcolor{violet}{\textbf{定义：}}若存在$\delta>0$，使得$U(M,\delta)\cap E=\varnothing$，则$M$为点集$E$的的\textbf{外点}。
+
+\textcolor{violet}{\textbf{定义：}}若对任意$\delta>0$，$U(M,\delta)$即有$E$内的点也有外的点，则$M$为点集$E$的\textbf{边界点}。
+
+\textcolor{violet}{\textbf{定义：}}$E$所有边界点的集合称为$E$的\textbf{边界}，记为$\partial E$。对于任意一个点集$E$与其余集$E^C$有公共边界，即$\partial E=\partial E^C$。
+
+\subsubsection{集合}
+
+\textcolor{violet}{\textbf{定义：}}设$E$为一个平面点集，若存在常数$\delta>0$，使得$E\subset U(O,\delta)$，则$E$为\textbf{有界集}，否则为\textbf{无界集}。
+
+\textcolor{violet}{\textbf{定义：}}若$E$中的每个点都是$E$的内点，则$E$为\textbf{开集}，若$E$的边界点都是$E$的点，则$E$为\textbf{闭集}。若一个点集是开集，则其余集为闭集，若一个点集为闭集，则其余集为开集。
+
+\textcolor{violet}{\textbf{定义：}}若$E$中任意两点，都可用一条完全属于$E$的曲线将其两点连接，则$E$为\textbf{（道路）连通集}，连通的开集为\textbf{开区域}，一个开区域和其边界点的并集为\textbf{闭区域}，统称\textbf{区域}。
+
+\textcolor{violet}{\textbf{定义：}}若$E$内任意一条\textbf{简单闭曲线}的内部还在$E$内，则$E$为\textbf{单连通区域}，否则为\textbf{多连通区域}。
+
+\subsubsection{聚点}
+
+\textcolor{violet}{\textbf{定义：}}对一个平面点集$E$，$M_0$为平面上一点，若对任意$\delta>0$，总有$\mathring{U}(M_0,\delta)\cap E\neq\varnothing$，即$M_0$的任意邻域中都含有异于$M_0$的$E$中的点，则$M_0$为$E$的\textbf{聚点}。
+
+\textcolor{aqua}{\textbf{定理：}}非空开集的内点余边界点都是这个点集的聚点，闭区域的任意一点都是其聚点。
+
+\textcolor{violet}{\textbf{定义：}}若存在$\delta>0$，使得$U(M_0,\delta)\cap E=\{M_0\}$，即如果$M_0$的某一邻域与点集$E$的交集是一个孤立的点$M_0$，则称$M_0$为$E$的\textbf{孤立点}。边界点要么是聚点要么是孤立点。
+
+\subsection{极限}
+
+对于一元函数的极限可用列举法，从两端逼近该点取极限，但是对于多元函数所处的邻域，逼近方向为无穷，所以不可能再通过取两个方向逼近的方式求极限。
+
+从点集来看\textcolor{violet}{\textbf{定义：}}设二元函数$f(P)=f(x,y)$的定义域为$D$，$P_0(x_0,y_0)$为$D$聚点。若存在常数$A$，对于任意给定正数$\epsilon$，总存在正数$\delta$，使得当$P(x,y)\in D\cap\mathring{U}(P_0,\delta)$时，都有$\vert f(x,y)-A\vert<\epsilon$成立，则常数$A$为$f(x,y)$当$(x,y)\rightarrow(x_0,y_0)$时的极限，记为$\lim\limits_{(x,y)\to(x_0,y_0)}f(x,y)=A$或$f(x,y)\to A((x,y)\to(x_0,y_0))$。
+
+如$\because xy\neq0$排除$xy$轴：$\lim\limits_{(x,y)\to(0,0)}\dfrac{\sqrt{xy+1}-1}{xy}=\lim\limits_{(x,y)\to(0,0)}\dfrac{xy+1-1}{xy(\sqrt{xy+1}+1)}$\\$=\lim\limits_{(x,y)\to(0,0)}\dfrac{1}{\sqrt{xy+1}+1}=\dfrac{1}{2}$。\medskip
+
+从邻域来看\textcolor{violet}{\textbf{定义：}}若二元函数$f(x,y)$在$(x_0,y_0)$的去心邻域内有定义，且$(x,y)$以任意方式趋向$(x_0,y_0)$时，$f(x,y)$均趋向于$A$，则$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f(x,y)=A$。
+
+根据邻域的定义，由于函数$\lim\limits_{(x,y)\to(0,0)}\dfrac{\sqrt{xy+1}-1}{xy}$在坐标轴上无定义，则极限不存在。
+
+此时两种定义就会有两种结论，所以为了避免这种定义不同的矛盾，就只会出现哪种定义下极限存在或都不存在的函数，如$\lim\limits_{\substack{x\to0\\y\to0}}(x^2+y^2)\sin\dfrac{1}{x^2+y^2}=0$。
+
+从现实角度来看，点集定义是更合理的，若要求一根弯曲铁丝在某点的导数，第二种定义无法求，所以不合理。而第二种定义是从一元极限定义直接升级过来，所以有一定局限性。
+
+\subsection{连续}
+
+\textcolor{violet}{\textbf{定义：}}若$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f(x,y)=f(x_0,y_0)$则称$f(x,y)$在点$(x_0,y_0)$处连续。
+
+若不连续，则不讨论间断类型。
+
+\subsection{偏导数}
+
+当含有两个以及三个变量时，若求一个极限，则有多个变量同时趋向，所以多个变量同时在变。为了运算简单，就假定只有一个变量在变，其他变量固定，从而直接降低成一元变量，只对一个变量求导，从而就是偏导数。
+
+\textcolor{violet}{\textbf{定义：}}设函数$z=f(x,y)$在点$(x_0,y_0)$的某邻域内有定义，若极限\\$\lim\limits_{\Delta x\to0}\dfrac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$存在，则称此极限为函数$z=f(x,y)$在点$(x_0,y_0)$处对$x$的\textbf{偏导数}，记为$\dfrac{\partial z}{\partial x}\bigg|_{\substack{x=x_0\\y=y_0}}$，$\dfrac{\partial f}{\partial x}\bigg|_{\substack{x=x_0\\y=y_0}}$，$z'\bigg|_{\substack{x=x_0\\y=y_0}}$或$f'_x(x_0,y_0)$。\medskip
+
+$f'_x(x_0,y_0)=\lim\limits_{\Delta x\to0}\dfrac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}=\lim\limits_{x\to x_0}\dfrac{f(x,y_0)-f(x_0,y_0)}{x-x_0}$。
+
+$f'_y(x_0,y_0)=\lim\limits_{\Delta y\to0}\dfrac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y}=\lim\limits_{y\to y_0}\dfrac{f(x_0,y)-f(x_0,y_0)}{y-y_0}$。
+
+\textcolor{violet}{\textbf{定义：}}若函数$z=f(x,y)$在区域$D$内的偏导数$f_x'(x,y)$、$f_y'(x,y)$仍具有偏导数，则其偏导数为函数$z=f(x,y)$的\textbf{二阶偏导数}。按照求导次序不同，有如下四个二阶偏导数。
+
+$\dfrac{\partial}{\partial x}\left(\dfrac{\partial z}{\partial x}\right)=\dfrac{\partial^2z}{\partial x^2}=f''_{xx}(x,y)$，$\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial y}\right)=\dfrac{\partial^2z}{\partial y^2}=f''_{yy}(x,y)$，
+
+$\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial x}\right)=\dfrac{\partial^2z}{\partial x\partial y}=f''_{xy}(x,y)$，$\dfrac{\partial}{\partial x}\left(\dfrac{\partial z}{\partial y}\right)=\dfrac{\partial^2z}{\partial y\partial x}=f''_{yx}(x,y)$。
+
+其中$f''_{xy}(x,y)$和$f''_{yx}(x,y)$为\textbf{混合偏导数}。二阶以及以上的偏导数均为\textbf{高阶偏导数}。
+
+\subsection{全微分}
+
+\textcolor{violet}{\textbf{定义：}}若函数$z=f(x,y)$在点$(x,y)$的全增量$\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)$可表示为$\Delta z=A\Delta x+B\Delta y+o(\rho)$，其中$\rho=\sqrt{(\Delta x)^2+(\Delta y)^2}$，$AB$不依赖$\Delta x$，$\Delta y$而仅与$x,y$相关，则称函数$z=f(x,y)$在点$(x,y)$可微，而称$A\Delta x+B\Delta y$为函数$z=f(x,y)$在点$(x,y)$的\textbf{全微分}，记为$\textrm{d}z$。
+
+$\textrm{d}z=A\Delta x+B\Delta y=\dfrac{\partial z}{\partial x}\Delta x+\dfrac{\partial z}{\partial y}\Delta y=\dfrac{\partial z}{\partial x}\textrm{d}x+\dfrac{\partial z}{\partial y}\textrm{d}y$。
+
+判断可微的步骤：
+
+\begin{enumerate}
+    \item 写出全增量$\Delta z=f(x_0+\Delta x,y_0+\Delta y)-f(x_0,y_0)$。
+    \item 写出线性增量$A\Delta x+B\Delta y$，$A=f_x'(x_0,y_0)$，$B=f_y'(x_0,y_0)$。
+    \item 写出极限$\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\dfrac{\Delta z-(A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$，若极限等于0，则$z=f(x,y)$在点$(x_0,y_0)$可微，否则不可微。
+\end{enumerate}
+
+% \textbf{例题：}已知函数$z=f(x,y)$的全微分$\textrm{d}z=2x\,\textrm{d}x+\sin y\,\textrm{d}y$且$f(1,0)=2$，求$f(x,y)$。
+
+% 解：
+
+\subsection{偏导数连续性}
+
+对$z=f(x,y)$，讨论其在某特殊点$(x_0,y_0)$处偏导数是否连续的步骤：
+
+\begin{enumerate}
+    \item 用定义法求$f_x'(x_0,y_0)$，$f_y'(x_0,y_0)$。（求某点偏导数）
+    \item 用公式法求$f_x'(x,y)$，$f_y'(x,y)$。（求偏导函数）
+    \item 计算$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_x'(x,y)$，$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_y'(x,y)$。（偏导函数求极限）
+    \item 若$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_x'(x,y)=f_x'(x_0,y_0)$，$\lim\limits_{\substack{x\to x_0\\y\to y_0}}f_y'(x,y)=f_y'(x_0,y_0)$若成立则连续，否则不连续。
+\end{enumerate}
+
+\textbf{例题：}设$z=f(x,y)=\left\{\begin{array}{ll}
+    (x^2+y^2)\sin\dfrac{1}{\sqrt{x^2+y^2}}, & x^2+y^2\neq0 \\
+    0, & x^2+y^2=0
+\end{array}\right.$，则四个结论中正确的个数为()。
+
+\ding{172}$f(x,y)$在$(0,0)$处连续。\qquad\ding{173}$f'_x(0,0)$，$f'_y(0,0)$存在。
+
+\ding{174}$f_x'(x,y)$，$f_y'(x,y)$在$(0,0)$处连续。\qquad\ding{174}$f(x,y)$在$(0,0)$可微。
+
+$A.1$\qquad$B.2$\qquad$C.3$\qquad$D.4$
+
+解：$\lim\limits_{\substack{x\to0\\y\to0}}(x^2+y^2)\sin\dfrac{1}{\sqrt{x^2+y^2}}=0=f(0,0)$。所以$A$正确。
+
+$f_x'(0,0)=\lim\limits_{\Delta x\to0}\dfrac{f(0+\Delta x,0)-f(0,0)}{\Delta x}=\lim\limits_{\Delta x\to0}\dfrac{(\Delta x)^2\sin\dfrac{1}{\sqrt{(\Delta x)^2}}-0}{\Delta x}=$\\$\lim\limits_{\Delta x\to0}(\Delta x)\sin\dfrac{1}{\vert\Delta x\vert}=0$。同理$f'_y(0,0)=0$。
+
+判断连续性，首先计算偏导数值，之前计算过：$f_x'(0,0)=f_y'(0,0)=0$；然后求偏导函数$f_x'(x,y)=2x\sin\dfrac{1}{\sqrt{x^2+y^2}}+(x^2+y^2)\cos\dfrac{1}{\sqrt{x^2+y^2}}\left(-\dfrac{1}{2}\right)\dfrac{2x}{\sqrt{(x^2+y^2)^3}}$\\$=2x\sin\dfrac{1}{\sqrt{x^2+y^2}}-\dfrac{x}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$，同理得$f_y'(x,y)=2y\sin\dfrac{1}{\sqrt{x^2+y^2}}$\\$-\dfrac{y}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$；最后一步查看偏导函数值与偏导数值是否相等，$\because\lim\limits_{\substack{x\to0\\y\to0}}2x\sin\dfrac{1}{\sqrt{x^2+y^2}}=0$，且$\lim\limits_{\substack{x\to0\\y\to0}}\dfrac{x}{\sqrt{x^2+y^2}}\cos\dfrac{1}{\sqrt{x^2+y^2}}$震荡，所以总的来说极限值不存在，就不会等于偏导数值，同理可得函数的偏导数在该点不连续。
+
+要求一个函数在某点可微，首先$\Delta z=f(0+\Delta x,0+\Delta y)-f(0,0)=[(\Delta x)^2+(\Delta y)^2]\sin\dfrac{1}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$。然后$A\Delta x+B\Delta y=f_x'(0,0)\Delta x+f_y'(0,0)\Delta y=0$。最后求极限$\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\dfrac{\Delta z-(A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}=\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\sqrt{(\Delta x)^2+(\Delta y)^2}\sin\dfrac{1}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$\\$=0$，所以在此点可微。
+
+综上正确的结论有\ding{172}\ding{173}\ding{175}三个，所以选$C$。
+
+\section{多元函数微分法则}
+
+\subsection{链式求导法则}
+
+主要对显函数的微分。
+
+多元函数链式求导法则与一元函数的求导法则类似。都是从因变量从中间变量走到自变量。一条路径是一个加项，多少条从因变量到所有自变量的路就有多少个加项。每条路上由不同的路段组成，若有$n$层中间变量，则有$n+1$路段，路段之间项是乘积形式，若变量只与一个变量有一条路，则是导数$\textrm{d}$，若一个变量到多个变量有多条路，则是偏导数$\partial$。
+
+\begin{minipage}{0.65\linewidth}
+    因变量$z$到$x$一共有两条路，所以两个和项。每条路都有两端，所以和项中有两个乘项。$z$到$uv$两个中间变量，所以是两个偏导$\dfrac{\partial z}{\partial u}$和$\dfrac{\partial z}{\partial v}$。$uv$都只有一条路直接连通$x$，所以都是导数$\dfrac{\textrm{d}u}{\textrm{d}x}$和$\dfrac{\textrm{d}v}{\textrm{d}x}$。一条路的每个路段的项相乘：$\dfrac{\partial z}{\partial u}\dfrac{\textrm{d}u}{\textrm{d}x}$和$\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。最后将每条路段相加：$\dfrac{\textrm{d}z}{\textrm{d}x}=\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。
+\end{minipage}
+\hfill
+\begin{minipage}{0.25\linewidth}
+    \begin{tikzpicture}[scale=1]
+        \filldraw[black] (-0.25,0) node{$z$};
+        \draw[black](0,0) -- (1,1) node[right]{$u$};
+        \draw[black](0,0) -- (1,-1) node[right]{$v$};
+        \draw[black](1.5,1) -- (2.5,0) node[right]{$x$};
+        \draw[black](1.5,-1) -- (2.5,0);
+    \end{tikzpicture}
+\end{minipage} \medskip
+
+\begin{minipage}{0.25\linewidth}
+    \begin{tikzpicture}[scale=1]
+        \filldraw[black] (-0.25,0) node{$z$};
+        \draw[black](0,0) -- (1,1) node[right]{$u$};
+        \draw[black](0,0) -- (1,0) node[right]{$v$};
+        \draw[black](0,0) -- (1,-1) node[right]{$w$};
+        \draw[black](1.5,1) -- (2.5,1) node[right]{$x$};
+        \draw[black](1.5,0) -- (2.5,1);
+        \draw[black](1.5,-1) -- (2.5,-1) node[right]{$y$};
+        \draw[black](1.5,1) -- (2.5,-1);
+    \end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}{0.65\linewidth}
+    因为因变量$z$到自变量$x,y$有较多条路径，所以分开分析。
+
+    对于$x$，有$z-u-x$，所以这条路为$\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}$，还有一条$z-v-x$，由于$v$只与$x$连通，所以是导数，该路为$\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$，所以$\dfrac{\partial z}{\partial x}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial z}{\partial v}\dfrac{\textrm{d}v}{\textrm{d}x}$。、
+
+    同理对于$y$，有$z-u-y$和$z-w-y$，且$u$有两条出路，$w$只有一条，所以$u$偏导，$v$导数，$\dfrac{\partial z}{\partial y}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial z}{\partial w}\dfrac{\textrm{d}w}{\textrm{d}y}$。
+\end{minipage} \medskip
+
+无论$z$对谁求导也无论求了几阶到，求导过后的新函数仍具有与原函数完全相同的复合结构。
+
+\textbf{例题：}设$z=f(e^x\sin y,x^2+y^2)$，其中$f$具有二阶连续偏导数，求$\dfrac{\partial^2z}{\partial x\partial y}$。
+
+解：$\because\dfrac{\partial^2z}{\partial x\partial y}=\dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial x}\right)$，$\therefore\dfrac{\partial z}{\partial x}=f_1'\cdot e^x\sin y+f_2'\cdot2x$。
+
+在求偏导时，将第一个中间变量记为$f_1$即之前的$u$，第二个中间变量记为$f_2$即之前的$v$。记$f$对$u$求偏导为$f_1'$，对$v$求偏导为$f_2'$同理二阶导也如此，下标为求导顺序。
+
+$\dfrac{\partial^2z}{\partial x\partial y}=\dfrac{\partial(f_1'\cdot e^x\sin y)}{\partial y}+\dfrac{\partial(f_2'\cdot2x)}{\partial y}$。
+
+其中$\dfrac{\partial(f_1'\cdot e^x\sin y)}{\partial y}=\dfrac{\partial f_1'}{\partial y}\cdot e^x\sin y+f_1'\cdot e^x\cos y$。所以难点就是$\dfrac{\partial f_1'}{\partial y}$。
+
+求导路径$f_1'-1-y$和$f_1'-2-y$：$=(f_{11}''e^x\cos y+f_{12}''2y)\cdot e^x\sin y+f_1'\cdot e^x\cos y$。
+
+\subsection{隐函数存在定理}
+
+主要对隐函数的微分。
+
+\section{多元函数极值最值}
+
+\subsection{概念}
+
+\subsection{无条件极值}
+
+\subsubsection{隐函数}
+
+\subsubsection{显函数}
+
+\subsection{条件极值与拉格朗日乘数法}
+
+\subsubsection{闭区域边界最值}
+
+\subsubsection{闭区域上最值}
 
 \end{document}
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